Some Notes on Pseudo-closed Sets

Pseudo-intents (also called pseudo-closed sets) of formal contexts have gained interest in recent years, since this notion is helpful for finding minimal representations of implicational theories. In particular, there are some open problems regarding complexity. In our paper, we compile some results about pseudo-intents which contribute to the understanding of this notion and help in designing optimized algorithms. We provide a characterization of pseudo-intents based on the notion of a formal context’s incrementors. The latter are essentially non-closed sets which – when added to a closure system – do not enforce the presence of other new attribute sets. In particular, the provided definition is non recursive. Moreover we show that this notion coincides with the notion of a quasi-closed set that is not closed, which enables to reuse existing results and to formulate an algorithm that checks for pseudo-closedness. Later on, we provide an approach for further optimizing those algorithms based on a result which correlates the set of pseudo-intents of a formal context with the pseudo-intents of this context’s reduced version.